91Ó°ÊÓ

Solving linear systems involves finding the values of variables that satisfy all equations in a given system. In this exercise, we need to find the values of \(u, v, w\) which satisfy three linear equations simultaneously. The key is to simplify the equations first. This involves eliminating any fractions and combining like terms to make calculation easier.
We start by multiplying each equation by the least common multiple (LCM) of the denominators to clear the fractions. Next, distribute any constants and combine like terms to simplify equations further. This process reduces complex equations into simpler forms, making it easier to isolate and solve for variables. By following these steps systematically, we can solve the systems efficiently.

Simultaneous equations are sets of equations that share common variables. In our example, we deal with three simultaneous equations that include variables \(u, v, w\).
We need to find values for these variables that satisfy all the equations at once.
By solving simultaneous equations, we essentially find the intersection points of these equations where they all hold true. This process requires a methodical approach, often beginning with simplifying and manipulating the equations to make variable isolation possible.
We use techniques like substitution and elimination to reduce the number of variables step by step, until we can find their exact values. Once we have the values, we verify them by plugging them back into the original equations. This confirms whether our solutions are correct.

Algebraic manipulation is crucial in solving linear systems. It involves rearranging and simplifying equations to make the solution process easier. We first clear out fractions by multiplying through by the LCM of the denominators.
Next, distribute constants and combine like terms, which means grouping similar terms together. This helps to isolate the variable terms from the constants. For example:
\[4(u+1) - 6(v-1) + 3(w+3) = 84\] simplifies to:
\[4u - 6v + 3w + 19 = 84\] Then, we subtract constant terms from both sides, like this:
\[4u - 6v + 3w = 65\].
This kind of manipulation strips down the equations to their simplest forms, paving the way for easier variable elimination.

Variable elimination helps to solve a system of linear equations by removing one variable at a time. This is done through subtracting or adding equations in a way that cancels out one variable.
For instance, consider two equations:

• \[2u + 2v + 3w = 123\]
• \[4u - 6v + 3w = 65\]

We subtract the first from the second to eliminate \w\:
\[4u - 6v + 3w - (2u + 2v + 3w) = 65 - 123\]
This simplifies to:
\[2u - 8v = -58\] which reduces further to:
\[u - 4v = -29\].
Now, we express \u\ in terms of \v\:
\[u = 4v - 29\].
Substitute this back into another equation and continue the process to isolate each variable. By systematically eliminating variables, we eventually find their exact values.